Gamma-Value Bunching

On the page dealing with sequences of eclipses I described how I discovered that eclipse gamma values (the measure of how far the centre-line of the eclipse passes north or south of the centre of the Earth) do not occur evenly but with very distinct preferred values which vary from one century to the next. I gave a suggested mechanism by which this bunching could occur but without much detailed justification that this would in fact result in exactly the observed effect. Time to dig deeper!

The first issue I had to investigate was that it wasn't clear that any bunching caused by one particular eclipse sequence would not be undone by a different set of bunches due to another sequence, or even that the proposed mechanism would indeed produce bunches in every case. Before describing this work I shall have to define some terminology, for the sake of brevity. A shorthand way of describing a sequence is by the number of 177, 148 and 29-day intervals it contains. For example, the oft-mentioned 177x5 148 29 148 29 148 would be 532. Analysis of actual eclipses shows that, ignoring oddities, there are in fact only five basic sequences of intervals which, in this terminology, are 710, 621, 610, 532 and 521: I shall call these "cycles". For each cycle one can calculate its "offset" (the window-distance between the first eclipse in the sequence and the last) and its duration. It turns out that there are only two different offset values (+1.51deg and -2.51deg) and two durations (3.800yrs and 3.315yrs), with cycles having the positive offset having the larger duration - 710, 621 & 532 fall into this category, while 610 & 521 are in the other. Now note that five times +1.51 plus three times -2.51 gives an almost zero result (+0.04 when done accurately) and that the same sum done with the corresponding durations comes to 28.94yrs. This means that after eight properly chosen interval cycles the last eclipse will be almost exactly back to where the first one started, and this will take only fractionally less than a whole number of years. In other words, after what one might call a "super cycle", the position of eclipses within the window and the date on which they occur will be repeated almost exactly. This means that any bunching of gamma values established in one super-cycle will tend to be reinforced in subsequent periods, not averaged out as one might imagine, resulting in the appearance of preferred values as observed. The different cycles within a super-cycle can't be chosen arbitrarily of course, as they must be realisable in practice. A typical super-cycle from the century 2300-2399 would be 532, 521, 532, 532, 610, 532, 532, 521 and from 2000-2099 would be 610, 710, 621, 610, 621, 610, 710, 621. Although both super-cycles satisfy the "five plus three" condition, note the quite different set of constituent cycles: this is due to the greater likelihood of "runs of 5" and 29-day events in the century at the maximum of the 565yr cycle. [After I had discovered this 28.94yr periodicity I did some further research and found that it was already known as a predictor of eclipses (in a similar way to the Saros period of 18yrs 11.3 days) and called an Inex. Its significance in the context of gamma-value distributions does not seem to have been documented, however]

Modelling

I wasn't confident I could derive an analytic treatment of the bunching phenomenon so decided to model it instead, using the two example super-cycle sequences given above. Initially, I kept things simple by using the cells of a spreadsheet as graph paper to accurately plot the relative positions of the middle (i.e. 177-day) eclipses in each sequence, in a similar way to the diagrams showing the progression of eclipses through the window of opportunity given earlier. Using a fairly simple model of lead/lag (max in April, min in October, simple linear variation in between) with the values quoted by Meeus (up to +/- 2deg), and an equally simple assumption that eclipses in a given sequence were all 6months apart, I then moved each of the eclipses by the appropriate amount and counted how many eclipses ended up in a given interval. To my considerable pleasure the eclipses on each diagram bunched up superbly, the "spread" of each bunch being no worse than the equivalent of 1.5 degrees! Repeating the exercise with lead/lag values of +/- 1.5deg and +/- 3deg produced similar results but the bunches were, in general, somewhat less well-defined for these values. This is almost certainly because a lead/lag value of 2deg is as close as possible to the offset values for each cycle (1.5 and 2.5 degrees) and so would tend to close up the spacings in an optimum way. The distance between the maxima of each bunch was, on each diagram and for every scaling factor, exactly 8 degrees - precisely as found from the plots of true gamma-values. Finally, the position of the bunches for 2000-2099 was not the same as those for 2300-2399, showing that the different sequence of cycles does indeed have an effect.

Encouraged by this success but keen to ensure that my simplifications had not given a false result, I wanted to extend the number of eclipses to include every eclipse in each sequence and work with lead/lag values that more accurately reflected the true intervals between eclipses. This required a "proper" spreadsheet, but at least it enabled me to vary the parameters of the model more easily and rapidly. It also allowed me to convert between window-position and equivalent gamma value more simply, thus allowing direct comparisons with real-life to be made. I again took the actual "super-sequences" given above, calculated where in the window and when each eclipse should occur (based on the nominal 177, 148 and 29-day differences) and applied a lead or lag value appropriate for the date, derived from a more realistic sine-curve variation but still assuming a maximum value of +/- 2deg.

When the results were plotted as a distribution of values, in the same way as those for the true gamma values, the final result was very good - see the graphs on the right (2000-2099 at top, 2300-2399 at bottom). The peaks in the distribution were in exactly the right places and even the behaviour at large values of gamma was well demonstrated. The only real difference was that the precise shapes of the peaks and troughs in the modelled distribution were not correct - they were rather too "mathematically perfect". As noted above in connection with the simplified model, one way to alter the shape of the peaks would be to use a larger maximum value of lead/lag. This tends to spread out each bunch, often asymmetrically. I found that if I increased the maximum the peaks were much closer to the required shape, even if still not entirely accurate, with the lead/lag value giving the best fit being around 3 degrees: click or tap the graphs to see the improvement. But what was the factor which was causing the lead/lag effect to need to be greater than the +/- 2deg stated by Meeus?

The influence of the Moon

The answer lies in the fact that the Moon's orbit is not circular. When it is far from the Earth (at apogee) it moves more slowly and so the line-up of Sun-Moon-Earth necessary for a new Moon will happen later than average i.e. the lunar month will be longer than average. Conversely, when it is near the Earth (at perigee) it will move faster, needing less time to get into position, and so the lunar month will be shorter than average. A graph of this variation is "sine curve" shaped, so because the values of points on such a curve tend to cluster about the maximum and minimum values (as there is less difference between each point here as compared to the sloping sections in-between) there will tend to be runs of several long or short lunar months. If these runs sit in the interval between two eclipses the overall time difference can thus vary considerably. This is born out by actual numbers - in the two centuries we have been considering so far, the range is 178.33 to 176.10 days (compared to the mean of 177.18). While not a lot in percentage terms, there is a disproportionate effect on the distance moved through the window of opportunity as all the extra (or lesser) time is spent within the window. In fact, with the given values, this ranges from 5.22 to 2.90 degrees (compared to the mean of 4.02) or a difference of +1.20 and -1.13 degree - yes, the magnitudes of the values really are different! Addition of these numbers to the +/- 2deg lead/lag figure (actually 1.914 in 2000AD when calculated accurately) gives overall values in agreement with the number found from the distribution graphs.

As with all things lunar, however, the situation isn't quite as clear-cut as just adding the two numbers would imply because there are actually four processes going on at the same time! The basic lead/lag effect caused by the eccentricity of the Earth's orbit is quite straightforward: max in April, min in October, period exactly 1yr. However, because the perigee/apogee points slowly move round the orbit (in the same way that the nodes do, if you can remember that far back in the discussion!) with a period of 8.85yrs, the additional effect of the Moon's orbital eccentricity has a period of not 1yr but almost 412days. The third factor (one I haven't mentioned so far) is a sort of combination of the above two - because the instant of new Moon is determined by the motion of both Moon and Earth, the variation of the lunar month due to the changing speed of the Moon (caused by the eccentricity of its orbit) is itself affected by a similar effect due to the eccentricity of the Earth's orbit. As with basic lead/lag, this has a period of exactly 1yr but has its maximum effect in early January (when the Earth is nearest to the Sun and hence moving fastest) and its minimum in early July.

Variations in the lunar month

Clearly, the maximum effect from the second and third processes will be when the Earth is moving fastest at the same time as the Moon is moving most slowly i.e. when lunar apogee is in early January. This will give a "maximally long" lunar month. Conversely, a maximally short lunar month will be produced by the opposite set of circumstances: lunar perigee in early July. One can also define "shortest long" and "longest short" months i.e. months where the two effects are at their maximum extent individually but acting in opposite directions - for example lunar perigee in January. Meeus states that the durations in these four cases are 29.83, 29.27, 29.65 and 29.43 days respectively. Given that the overall average is 29.53 days, a bit of maths soon shows us that the maximum influence of the perigee/apogee effect alone is +0.211/-0.178 days and for the Earth's orbital eccentricity alone is +0.091/-0.078 days. As mentioned above, the effect is actually slightly greater in the "long" direction than the short: this may be because the very largest values of apogee (when the Moon is at its very slowest) and very smallest values of perigee (when it is fastest) both occur when the Earth/Moon system is nearest to the Sun i.e. around January. This is to the benefit of longer lunations but not shorter (which would require smallest perigee values in July).

Taking into account the different periodicities of the two effects, calculation shows that an eclipse season made up of six months centred on a maximally-long month would last 178.7 days and one centred on a maximally short month 177.3 days. Unfortunately, these are clearly not the values we are looking for! The answer lies in the fourth effect: because the perigee/apogee points move round the Moon's orbit in the same direction as the Moon itself, and at constant speed, when the Moon is moving more slowly (at apogee) they get further ahead than average and when it is moving faster (at perigee) they get less far ahead. This means that new Moon is in alignment with lunar apogee for a much lesser time than with lunar perigee (as can be seen in tabulations of these things on the Internet), and so the lunar months adjacent to a very long one are not as long as one might think but those adjacent to a very short month are shorter. This effect slightly decreases the duration of a maximally-long season and greatly decreases that of a maximally-short one.

While it is easy to "turn off" one effect or another on a spreadsheet in order to examine it in isolation it's a little more difficult in real life! To determine the magnitude of the fourth effect I thus plotted the difference of each lunation from the average for the period across a time when new Moon was close to lunar apogee in January i.e when maximally-long lunations would be produced (2088/89, in fact). I then set my model for the same circumstances, but with a parameter which would vary the alignment periods described above, and also plotted the results. Both plots were asymmetric with respect to long and short lunations, as expected. With the parameter set to give the same degree of asymmetry on the actual and calculated graphs, the durations of the longest possible and shortest possible eclipse seasons were to 178.37 and 176.08 days respectively. Comparing these values to the actual maximum and minimum duration "177-day" events for my two test centuries given above (178.33/176.10) shows an excellent agreement, thus confirming that the processes are indeed acting in the way described.

A complete theory

So, now the various mechanisms have been identified and the magnitudes of their effects calculated, does adding them into the calculations of gamma-value distributions actually make any difference? Gratifyingly, yes it does! Once the correct time relationship between the various factors is correctly represented, the "actual" and "calculated" graphs sit very nearly exactly on top of one another: see the graphs on the right. I thus feel I can claim with a good degree of confidence that I have shown that the emergence of preferred values of gamma is due to the bunching of the positions of eclipses within the window of opportunity due to a) the lead/lag effect caused by the variable speed of the Earth in its orbit and b) variations in the length of lunar months caused by the variable speeds of both the Moon and the Earth, and the precession of the line of apogee/perigee.

Eclipse regression

There is perhaps one final item of interest to take from all this - given that the shift in position caused by all the factors acting together is something over +/- 3 degrees and that the spacing of eclipses if unperturbed is 4 degrees, a pair of eclipses affected to the maximum extent but in opposite directions would in fact "swap places" within the window of opportunity! (and a bit more). That this actually happens can be seen by examining the eclipses in 2088/89. These occur in April and October in both years - the optimum time for all the above mechanisms to add up as long as lunar apogee occurs in January 2089, which of course it does (hence the reason for using this period in the calculation above). In each year, the October eclipse has a gamma value which is smaller than that for the April event despite all of them being "177-day" eclipses (which should have steadily increasing values). This means that on a diagram such as I used earlier the October eclipses would be placed before the April ones, not after as one would expect. The other side of this coin is that the April eclipses have gamma values very much larger than the preceding October ones as these pairs of eclipses have moved far apart. In terms of the position of the eclipses on the Earth's surface, instead of successive tracks moving smoothly from south to north the October ones actually move backwards i.e. are further south than the preceding April ones, while April ones leap northwards rather than just slowly moving up. In fact, the apogee/perigee alignment is still sufficiently good in 2090 that the eclipse of 23rd September, which happens just 5hrs after the exact time of lunar perigee, is shifted sufficiently to move it out of the partial eclipse band into the total eclipse sector. Thus instead of the usual run of partials, then totals or annulars, then partials again as eclipses move through the window of opportunity, there's this odd total completely out of sequence! To illustrate the above points, the table below compares this period with the time 4yrs previously when all lead/lag factors were near their mean values: apogee alignment in July and eclipses in December and June [As ever, the sign of the gamma values, and therefore the N/S labels of the latitudes, has been adjusted to that of a "descending node" eclipse to enable easier comparisons to be made].

The greatest positive and negative gamma differences for "177-day" eclipses in the century 2000-2099 are +0.8872 and -0.1925. Assuming the average difference (0.3431) corresponds to the standard 4.02deg spacing, the extremes correspond to 10.39 and -2.25deg i.e. differences of +6.37 and -6.27deg from the mean value. Given that pairs of eclipses are involved in the bunching process, this means that each of them moves an average of +/- 3.16deg, which is in good agreement with the previously mentioned figures for total lead/lag.

Just to complete the circle of this investigation, it is possible to see the periods of negative gamma difference on the plots of gamma value and cumulative timing difference shown right at the beginning. This is very obvious on the upper graph (2000-2099) where the second and final "upslopes" have a sawtooth appearance. You have to look a little harder on the lower graph (2300-2399) but near to the beginning and the end there is a definite zig-zag as the blue graph crosses the x-axis. It is also interesting to compare how the "jaggedness" of each upslope (which is the area where the 177-day eclipses lie) varies across each graph as the various processes come into and out of alignment.

The middle eclipses of the sawtooth regions on the 2000-2099 graph are 31st March 2071 and 10th April 2089, which are separated by 18yrs 10days. On the 2300-2399 graph they are 29th April 2302 and 9th May 2320, also separated by 18yrs 10days. This is, of course, not a coincidence: those who have been diligently following these pages will recognise this as the Saros period. This is the interval after which all the circumstances of an eclipse repeat almost exactly, so it is no surprise that it will also produce sequences of "maximal shift" events: further ones will happen in the spring of 2017, 2035 and 2053 for example.

Well wasn't that fascinating?